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Abstract:
Based on a previous work handling the 4-particle amplituhedron at 3-loop, we
have found an extremely simple pattern, yet far more non-trivial than one might
naturally expect: the all-loop Mondrian diagrammatics. By further simplifying
and rephrasing the key relation of positivity in the amplituhedron setting,
remarkably, we find a completeness relation unifying all diagrams of the
Mondrian types for the 4-particle integrand of planar N=4 SYM to all loop
orders, each of which can be mapped to a simple product following a few plain
rules designed for this relation. The explicit examples we investigate span
from 3-loop to 7-loop level, and based on them, we classify the basic patterns
of Mondrian diagrams into four types: the ladder, cross, brick-wall and spiral
patterns. Interestingly, for some special combinations of ordered subspaces (a
concept defined in the previous work), we find failed exceptions of the
completeness relation which are called "anomalies", nevertheless, they
substantially give hints on the all-loop recursive proof of this relation.
These investigations are closely related to the combinatoric knowledge of
separable permutations and Schroeder numbers, and go even further from a
diagrammatic perspective. For physical relevance, we need to further consider
dual conformal invariance for two basic diagrammatic patterns in order to
obtain the correct numerator for an integrand that involves such patterns,
while the denominator encoding its pole structure, and also the sign factor,
are already fixed by rules of the completeness relation. With this extra
treatment to ensure the integrals are dual conformally invariant, Mondrian
diagrams can be exactly translated to their corresponding physical loop
integrals, after the summation over all ordered subspaces that admit each of
them.